Method of determining the polarization of a light beam



Jan. 22, 1952 MUELLER 2,583,186

METHOD OF D RMINING MPOLARIZATION OF A LIGHT Filed NOV. 19, 1949 2 SHEETS--SHEET l |||llllllllilllllllllll Jan. 22, 1952 H. MUELLER 2,583,186

METHOD OF DETERMINING THE POLARIZATION v OF A LIGHT BEAM Filed NOVfilQ, 1949 2 SHEETSSHEET 2 Patented Jan. 22, 1952 UNITED s PATENT err-ice METHOD OF DETERIWINING THE-POLARIZA- iZIIION' DF A LIGHTWBEAM Hans Mueller, Cambridge, "Masa, assignor, by mesne assignments, "to Research Corporation, N ewYork,*"-N. 'Y., a corporation offNew :Yor'k Application November 19, 1949,Serial"N0. 128,407

The present invention .relates to methods .and apparatus .fortheanalysis lofl'light, and more-particularly for the analysis of partially polarized light. .Such apparatus .is useful in various app1ications;.for example, in .thedetermination of the particle size distribution of a polydisperse smoke, as will be explainedhereinafter.

The term fipartially polarized light refers "to light which is .neither polarized nor natural. Polarized'lig'ht is light which has specific transverse properties. There is a variety of variously (polarized light beams, which can be categorized as plane polarized, circularly polarized, or felliptically :Dolarized. vNatural .light :is -.a random imixturex of all these various types. :ture of polarized components which is not-completely ran'domis known as ipartially polarized light. The various types of light can be considered as a continuum, with natural light at one extreme and polarized light at :the :other,

:andwith partially polarzied light-10f varying ;degrees of polarizationzlyingiin.between. flfhe"degree :of depolarization of .light beam is an index of its location in theeontinuum.

vThe object of the present invention is .to -.pro-

In ath'e accompanying diagrams, Figs. 51 fandfi v:are diagrams of the-apparatus .used inmaking ipart iOf the required measurements; .2Fi'g. 33 is :a diagram of the apparatus used in making other of the ..required.measurements; Pandiliigs. '14 :and 25 "are diagrams utorexplain' theioperation -T0f themresentiinvention.

.In rFig. 7'1 .the arrow 2H] represents-alight beam to :be analyzed. The beam willusually, though not necessarily, be monochromatic, :first :passes through a =bi-pol'arizer l 2.

The light This can be a Wollaston-, Rochonor .Senarmont-prism,

or a Polaroid "bipartite plate (also called a 'split field Polaroid). The biepolarizer splits the light beam into two plane polarizedfields oriented at right angles to each other. The line separating the fields is convenientlyrefen'ed to as the axis of the bi-polarizer. .The :light .then passes through the analyzer l3, which may be a Nicol prism or a Polaroid'plate. It then passes into the telescope l4.

As shown in Fig.-3, la-'quarter-wave plate I5 is,

"for certain "purposes, introduced ahead of the bi-polarizer. 1 i

ln 'theoperation ofthepresent invention, only three simple angular-measurements 'are required.

In the first measurement, "the axis of "the -bipolarizer is in a certain position, conveniently shown vertical. Looked at through'the'telescope Hi, "the "two "halves 'of the bi-polarizer *appear, in general, t'o 'beoi unequal brightness. 'The op- *erator rotates theanalyzer I 3 until both sections of the bi-polarizer appear to be equally bright. There are two positions of th'e analyzer inwhich this a condition is iulfilled. The operator deterinesthesetwo positions andmeasures' thean'gle of rotation :between them. fllh'i's an-glewillzberre .ierred to as a1.

xToperformlthe second measurement, as shown in Fig. 2, the operator turns the "bi-"polarizer 12 through an tangle of 45 in either-direction "from "the vertical. He .then rotates the analyzer as before :to vdetermine the two positions in "which the sections of thebiepolarizer apDearfto be-equal- .ly bright,=.and measures the angle between :these two positions. This angle will be referred to as as.

For :the :third measurement the quarter-wave plate idwith itsiaxishorizontaliis inserteclahead otf the -bi-polarizerand. analyzer, as inFigAS. .The axis' of the bi-polarizer 12 isset at the same posi- -tion=as in the-secondmeasurement. .Theanalyzer I3 :is .'-rotated as in the ,first two ..measurements, and the angle between the twopositionsof equal brightness is measured. This angle will be referred to as as.

The three angles thus measured are enough :to determine whether the light is polarized, partially polarized ornatural, according to the following simple rulezCompute the sum cos m-l-cos a2 -|-cos as. If this sum has the value 1 the lightis totally polarized; if it is smaller than 1 the light is partially polarized; and z-ifzit is zero the light is natural. The "deviation of the sum from 1 is a measure of the degree -of depolarization of "the light.

The operation of thepresent invention may be mostreadily explained-by a considerationof the .two -.limiting -cases--natural light and ,polarized light.

.The case of natural light is relatively simple. sineezit'has no specific transverse; properties, the light ,passed :through the .two fields :of the bigp'olarizer will be equal @111 intensity, for each of the three mea-surements. -As viewed through-the teleseopeatheatwo areasiwillaappear equally bright 'when the axis et the analyzerimakesza 415 tangle 3 with the axis of the bi-polarizer. There are obviously two positions of the analyzer in which this condition is fulfilled, and the angle between them is 90. All three anglesa1, a2, a3are therefore 90. The sum cos ar -i-cos a2 +CS 0.3 is, of course, zero.

The case of polarized light is more complex. The polarized beam can be considered in terms of its electric vector, which vibrates in the plane of the wave front. This vector can be resolved, in turn, into two components, a horizontal component (Ex) and a vertical component (Ey) In the case of monochromatic light, these vary sinusoidally. The frequencies of the two sinusoids are the same, but their magnitudes are in general different; and there is generally a phase difference, which is represented by the phase angle This analysis is shown diagrammatically in Fig. 4. The vectors E and Ex are represented by circles whose radii correspond to the magnitudes of the vectors. The instantaneous value of each vector is represented by the projection on the diameter of its circle of a point moving with constant velocity around the circumference. The Figures 1, 2, 3, around each circle represent corresponding positions of the moving points. The difference in phase is shown by the fact that the numbers on one circle are offset by the angle from the numbers on the other circle.

The combination of Ex and Ey usually gives an ellipse, as shown in Fig. 4. When the phase angle is zero or 180, the ellipse degenerates into a straight line. This represents a plane polarized beam. If the phase angle is 90, and EI=E the ellipse becomes a circle. This represents a circularly polarized beam.

If a bi-polarizer with its axis vertical is placed inthe path of the light beam represented by Fig. 4, the beam is split into two fields, one vertically polarized, and the other horizontally polarized. The intensities of the two portions are, respectively, E and Ex When viewed through the analyzer, these intensities are reduced by the factors cos 'y and sin 7 respectively, where Y represents the angle between the trans mission direction of the analyzer and the vertical. The two fields of the bi-polarizer will appear equally bright for two angles '7. These angles are of equal magnitude but opposite sign. Thus 27:11.1. Then from which 1 E tan 'y=tan oq= (1) and E E;- COS d1=m From the last equation, the geometric significance of a1 is apparent. It is the angle between the diagonals of the rectangle ABCD in Fig. 4, having horizontal and vertical sides tangent to the ellipse. This result permits an interpretation of the angle a2, obtained with the bi-polarizer at a 45 angle from the vertical. Since a rotation of the bi-polarizer means simply a rotation of the reference axes, a2 must be the angle between the diagonals of a rectangle whose sides are tangent to the ellipse and parallel to the +45 and ---45 directions. The geometrical construction leading to as is shown in Fig. 5. It

4 shows that for plane polarized light, for which 11:0, (12 is complementary to 11. In the other limiting case, when =90, the rectangle becomes a square, whence a2=90. Thus it is realized that 0.2 depends not only on Ex and Ey, but also on and the two limiting cases suggest that cos a2 is proportional to cos A simple computation shows indeed the validity of the equation To explain the significance of 0.3 the action of the quarterwave plate must first be explained. This may be a plate of mica of definite thickness or, more conveniently, a sheet of plastic material which has been stretched or rolled, with the result that plane polarized light with its electric vector along the direction of stretch travels with a different speed through the sheet from light with the vector normal to the stretch. Hence the Ex and B components of any beam change their phase difference during passage through the sheet. Thus the plate may be generally termed a phase-displacing device. The thickness of the sheet and the stretch are adjusted to give a 90 shift of phase. Since in the as determination the bi-polarizer has the same orientation as for the a2 measurements, 0.3 is given by the same formula as 0.2 if (+90) is substituted for 11. Hence COS 2E.. E, EH12. (4)

Since the values of cos a1, cos a2 and cos as have been calculated, the sum cos 0.1+COS2 a2-i-C0S 113 can now be determined. By squaring and adding (2), (3) and (4), it is readily seen that cos a sin The total intensity I may be determined by a photometric measurement, but such a measurement is unnecessary if only the state of polarization is required, since the ratios M /I, F/] and S /I are then sufiicient. If it is desired to solve for the parameters associated with elliptical polarization, the above equations can be solved for the ratio Ey/Ez and the phase angle Furthermore since elliptically polarized light may be considered as made up of plane polarized and circularly polarized components, the components themselves may be obtained in a manner which requires no explanation here but will be clear to those skilled in this art.

It has thus been shown that for naural light or 1) M +F"+S =0 while for totally polarized light cos m-i-cos az-i-cos (13:1

Since these are the two possible extremes, a measurement on partially polarized light, i. e., a mixture of natural and polarized light, will give a result in which the sum of the squares of cosines will be between zero and one. The tion of the sum irom one is then a measure of the depolarization oi" the light. In any event, the four Stokes parameters 1, *5 and serve to characterize the light comple' cly. (As in the special cases above described, it is not usually necessary to ow the actual magnitudes of these intensities since the quantities M/I, 3/1 as given directly by the measurements, will 5 lice. These quantities may be termed the re.a tive intensities or relative Stokes param te s From the parameters I, M, to determine t .e intensities of the coponents. in other words, it is possible to deter the remainder will. be e -.ptically polariz of certain eccentricity and phase; then 1 the elliptically polarized light may be into its linear and circular coniponc The relations among these various quan es, however, are not simple in the case of partialy polarized light, nor is it usually necessary to effect a resolution of intensities into natural polar ized components, since a knowledge or the Stokes parameters IV, F and S in relation to the total intensity I will suffice for most purposes. In event, the measurement of the Stokes pa ameters according to the present invention provides all necessary information as to the state of polarization of the light.

The importance of the Stokes parameters arise mainly from the fact that if a light beam with parameters 1, M, F, S is incident on any of optical instrument which transmits, reflects or scatters the light, the Stokes components I, M, F, S of any beam emitted therefrom are given by linear functions of the Stokes components or" the incident beam. If, therefore, the beams incident on and emitted from a certain instrument are measured, information as to the character oi the instrument may be obtained; for example, ternpered glass, because of certain anisotropic char-= acteristics, converts a polarized beam into a partially polarized beam, and measurements of the beams may be utilized to determine the efiectiveness of the tempering. As another example, the scattering of light-by particles suspended in a liquid or a gas may be considered. Polarized light is to some extent depolarized by the scattering effect of polydisperse suspensions, and the Stokes parameters of the beams emitted in various direc tions are significant in determining the size and distribution of the particles.

By utilizing the precise sequence of steps described herein, the results are attained directly in terms of the Stokes parameters. In general, however, the exact angles described above are not necessary. Thus, after measuring :11 with the axis of plate i2 vertical, it is not essential to set the axis of plate !2 to exactly 45 for the second step. Any angle B, not necessarily 45, will serve. Also, for the third step, the plate I2 may be set at any angle C,'and the quarter-wave plate at any angle D. From these values and the measured ill-values, the Stokes parameters may be derived, or the a-values themselves may be considered determinative. It will be understood that there may be certain degenerate values of B, C and D which will not give enough independent equations to solve for the necessary paramstore, but in general, any values of these angles may be used to provide a complete determination of the state of polarization of the lighf".

practice or" the invention, substantially mo; chromatic light will ordinarily be used, es-

in experiments on scattering which depends on the wavelength. However, so far as the procedure of measurement is concerned, it is not limited to monochromatic light, but may be carried out with polychromatic light within the range of: achromatism of the parts of the equipment. In this respect, the quarter-wave plate iii is usually the limiting factor, although such plates are available which are achromatic over a substantial portion of the spectrum.

The present invention therefore provides a simple easily operated method and apparatus for completely determining the state oi polarizatio of any light beam, whether natural, or paror completely p032 izecl.

....aving thus described the invention, I claim: 1. The method of determining the state of polarization of a beam of light which comprises iii-polarizing the li ht beam to obtain two planepolarized fields having their axes of polarization at an angle to each other, passing the lei-polarized beam through an analyzer, rotating the analyzer to' determine two positions in each of which the appear of equal intensities and measuring the angle betwe n said two positions, setting the axis of lei-polarization at a determinate angle from its initial position, again rotating the analyzer and measuring the angle between the two positions in which the fields appear of equal intensities, introducing a phase-displacing device, and again rotating the analyzer and measuring the angle between the two positions in which the fields appear of equal intensities, whereby the state of polarization is determined from the angles thus measured.

2. The method of determining the state of polarization of a beam of light which comprises introducing a bi-polarizer into the path of the light beam to obtain two plane-polarized fields having their axes of polarization at right angles to each other, passing the oi-polarized beam through an analyzer, rotating the analyzer to determine two positions in each of which the fields appear of equal intensities and measuring the angle between said two positions, setting the axis of bipolarization at an angle of 45 degrees from its initial position, again rotating the analyzer and measuring the angle between the two positions in which the fields appear of equal intensities, introducing a phase-displacing device while maintaining said bi-polarizer with its axis at said 45 degrees angle from its initial position, and again rotating the analyzer and measuring the angle between the two positions in which the fields appear of equal intensities, whereby the state of polarization is determined from the angles thus measured.

HANS MUELLER.

REFERENCES CITED The following references are of record in the file of this patent:

Theory of Light, by T. Preston, published by MacMillan & (30., London, second edition, 1895, pages 44, 296, 299, 403-418.

Principles of Optics, by A. C. Hardy et al., published by McGraw-Hill Book Co., New York, first edition, 1932, pages 595-618. 

